Recycling Queries in PCPs and in Linearity Tests

We study query-efficient Probabilistically Checkable Proofs (PCPs) and linearity tests. We focus on the number of amortized query bits. A testing algorithm uses $q$ amortized query bits if, for some constant $k$, it reads $qk$ bits and has error probability at most $2^{-k}$. The best known PCP construction for NP in this respect uses 3 amortized query bits (Hastad, STOC97); at least one amortized query bit is necessary, unless P=NP (Bellare, Goldreich and Sudan, FOCS95). This parameter is an extremely naturalone and has applications to proving non-approximability forconstraint satisfaction problems. Furthermore, a PCP characterizationof NP with less than 2 amortized query bits would imply a separationof the PCP model from the 2-Prover 1-Round model.

Our approach is to take an atomic verification procedure andthen iterate it several times, saving queries by recycling thembetween different iterations of the atomic test.

We first apply this idea in order to develop query-efficientlinearity tests. Linearity testing is a problem closely relatedto testing the Long Code and making PCP constructions. It is also asignificant combinatorial problem still lacking tight characterizations,except for the case of three queries (Bellare et al. FOCS95).The best known linearity test uses 3 amortized query bits (Bellareet al., FOCS95); a different one achieves 1 amortized free bit (a different parameter related to the Max Clique problem) but uses an unbounded number of amortized query bits (Bellare, Goldreich and Sudan, FOCS95). We develop a general analysis technique and a linearitytest achieving simultaneously amortized query complexity 1.5 and amortized free bit complexity 1/2. This test answers an open questionraised by Bellare, Goldreich and Sudan.

We then show how to adapt a weaker result to the PCP setting, and we obtain a PCP for NP that makes 5 queries and has error probability 1/4, so that its amortized query complexity is 2.5.